Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number.
Answer: If the length, width, and height of the rectangular prism are $a$, $b$, and $c$, then we are given $ab=48$, $bc=49$, and $ac=50$.  Since we are looking for $abc$, the volume of the rectangular prism, we multiply these three equations to find \begin{align*}
(ab)(bc)(ac)&=48\cdot49\cdot50 \implies \\
a^2b^2c^2&=48\cdot49\cdot 50 \implies \\
(abc)^2 &= 48\cdot49\cdot50 \implies \\
abc &= \sqrt{48\cdot49\cdot50} \\
&= \sqrt{(16\cdot 3)\cdot 7^2\cdot(2\cdot 5^2)} \\
&= 4\cdot7\cdot5\sqrt{2\cdot3} \\
&= 140\sqrt{6},
\end{align*} which to the nearest whole number is $\boxed{343}$ cubic units.